Math 2.0

Real functions Concept

Limit along different paths

Builds on Limit of a function of several variablesParametrised curve

Summary

Let ff be a function on D⊆ℝnD \subseteq \mathbb{R}^n and let aa be a limit point of DD. Let γ ⁣:(0,1]→D\gamma \colon (0, 1] \to D be a path, for example a curve, with γ(t)≠a\gamma(t) \ne a for each tt and γ(t)→a\gamma(t) \to a as t→0+t \to 0^+. If lim⁡x→af(x)=L\lim_{x \to a} f(x) = L, then lim⁡t→0+f(γ(t))=L\lim_{t \to 0^+} f(\gamma(t)) = L. So if two such paths give different limits, or one path gives no limit, then the limit of ff at aa does not exist.

Remarks

For f(x,y)=xy/(x2+y2)f(x, y) = xy / (x^2 + y^2) on ℝ2∖{(0,0)}\mathbb{R}^2 \setminus \{(0, 0)\}, the path y=mxy = m x gives the limit m/(1+m2)m / (1 + m^2) at the origin. This depends on mm, so ff has no limit at (0,0)(0, 0). Equal limits along all straight lines do not prove that the limit exists. For f(x,y)=x2y/(x4+y2)f(x, y) = x^2 y / (x^4 + y^2), each line through the origin gives the limit 00, but the parabola y=x2y = x^2 gives 1/21/2.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptParametrised curve, concept, no evidence levelParametrised curveconceptLimit along different paths, concept, no evidence levelLimit along different…concept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Function of several variables concept
  4. Vector-valued function concept
  5. Limit of a function of several variables concept
  6. Parametrised curve concept
  7. Limit along different paths concept

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Limit_of_a_function, Functions of more than one variable
About this page
Tag0BJ6H85DRQ
URIm2://0BJ6H85DRQ
Aliasestwo-path-test
Kindconcept
WritingW0 stub
MSC26B05
UsesLimit of a function of several variables, Parametrised curve

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by